In string theory, the value of the dark energy depends on the shape of the internal dimensions, as well as possible branes and/or magnetic-like fluxes that are present there.
The constructions that are simplest to analyze are those that have zero dark energy and have unbroken supersymmetry.
However, there has been vigorous ongoing research to construct more realistic configurations with positive values of the dark energy and broken supersymmetry, with one of the first plausible scenarios being [1]}. Several objections and counter objections to such scenarios have been raised, and the effort to make them more concrete is still ongoing. This involves the intricate mathematics of Calabi Yau manifolds, as well as sophisticated control of non-perturbative corrections. Other compactifications developed in parallel yield power law stabilization mechanisms, including more general curved manifolds which are also amenable to concrete mathematical analysis. These constructions indicate that string theory has a vast landscape of possible solutions, each of them with distinct four dimensional physics. There has been a significant effort at understanding the constraints on the effective four dimensional theories that come from such constructions, see [2]} for a recent review of some of these ideas.
As mentioned in the previous section, one rather generic feature of these constructions is the appearance of a possibly large number of axion-like fields. In addition to their relevance for particle physics, such particles are also useful for constructing inflationary models and may contribute a time-dependence to dark energy in the present universe.
